Download EE 321 Analog Electronics, Fall 2011 Homework #5 solution

EE 321 Analog Electronics, Fall 2011
Homework #5 solution
3.37. Find the parameters of a piecewise-linear model of a diode for which vD =
0.7 V at iD = 1 mA and n = 2. The model is to fit exactly at 1 mA and 10 mA.
Calculate the error in millivolts in predicting vD using a piecewise-linear model
at iD = 0.5, 5 and,
14 mA
vD
We have iD = IS e nVT , so
v
− nVD
IS = iD e
T
= 1 × 10−3 × e
−
0.7
2×25.2×10−3
= 0.93 nA
Next,
vD = nVT ln
iD
IS
= 2 × 25.2 × 10−3 × ln
10 × 10−3
= 0.82 V
0.93 × 10−9
Then we have
rD =
vD10 − vD1
0.82 − 0.7
=
= 13 Ω
iD10 − iD1
10 − 1
and
vD0 = vD1 − rD iD1 = 0.7 − 13 × 1 × 10−3 = 0.69 V
We can now calculate vD based on the piecewise linear model as
vD = vD0 + iD rD
or based on the exponential model
vD = nVT ln
iD
IS
and those are tabulated here
iD vD (exp) vD (lin) error
0.5
0.67
0.70
0.03
5
0.78
0.76
0.02
14
0.83
0.87
0.04
3.54. In the circuit shown in Fig. P3.54, I is a DC current and vs is a sinusoidal
signal. Capacitors C1 and C2 are very large; their function is to couple the signal
to and from the diode but block the DC current from flowing into the signal
source or the load (not shown). Use the diode small-signal model to show that
the signal component of the output voltage is
nVT
vo = vs
nVT + IRs
If vs = 10 mV, find vo for I = 1 ma, 0.1 mA, and 1 µA. Let Rs = 1 kΩ and n = 2.
At what value of I does vo become one-half of vs? Note that this circuit functions
1
as a signal attenuator whith the attenuation factor controlled by the value of the
DC current I.
The signal portion is transferred from input to output according to a voltage division between
RS and rD ,
vo = vs
rD
Rs + rD
where
dvD
=
rD =
diD iD =I
diD
dvD
−1
id =I
=
I
nVT
−1
=
nVT
I
and thus
vo = vs
nVT
I
RS +
nVT
I
=
nVT
IRs + nVT
Find vo for several values of I, and vs = 10 mV, Rs = 1 kΩ, and n = 2. It is tabulated below
I (mA) vs (mV)
1
0.48
0.1
3.4
10−3
9.8
Value of I for which vo = v2s :
1
nVT
=
nVT + IRs
2
2 × 25.2 × 10−3
nVT
=
= 5.0 × 10−5 A = 50 µA
I=
3
Rs
1 × 10
3.59 Consider the voltage-regulator crictuit show in Fig. P3.59. The value of R
is selected to obtain an output voltage Vo (across the diode) of 0.7 V.
2
(a) Use the diode small-signal model to show that the change in output voltage
corresponding to a change of 1 V in V + is
nVT
∆Vo
=
∆V +
V + + nVT − 0.7
This quantity is known as the line regulation and is usually expressed in
mV/V.
(b) Generalize the expression above to the case of m diodes connected in seris
and the value of R adjusted so that the voltage across each diode is 0.7 V
(and Vo = 0.7 m V).
(c) Calculate the value of line regulation for the case V + = 10 V (nominally)
and (i) m = 1, and (ii) m = 3. Use n = 2.
(a) We can write
iD =
and we are interested in finding
V
+
=R
vD
V + − vD
= IS e nVT
R
dvD
.
dV +
v
D
R
we can re-arrange
+ IS e
vD
nVT
and then compute
vD
RIS nV
dV +
=1+
e T
dvD
nVT
3
vD
= vD + RIS e nVT
Now, we are interested in evaluating this at a bias point vD = VD , for which
VD
ID = IS e nVT
so we can insert that and get
dV +
RID
=1+
dvD
nVT
at that bias point we have RID = V + − VD , so we can write
dV +
V + − VD
=1+
dvD
nVT
and we are asked to compute
1
nVT
dvD
=
=
+
D
dV +
nVT + V + − VD
1 + V nV−V
T
With the bias point VD = 0.7 V this is identical to the expression we were asked to
compute.
(b) In this case we just write
vD
V + − mvD
iD =
= IS e nVT
R
re-arrange
V
+
=R
mv
D
R
+ IS e
vD
nVT
vD
vO
= mvD + RIS e nVT = vO + RIS e mnVT
and compute
vO
vD
dV +
RIS nV
RIS mnV
=1+
e T =1+
e T
dvo
mnVT
mnVT
Now note that at the bias point, vD = VD =
VO
,
m
we can substitute
vD
ID = IS e nVT
RID
V + − VO
V + − mVD
dV +
=1+
=1+
=1+
dvo
mnVT
mnVT
mnVT
Finally we can compute
4
1
mnVT
dvo
=
=
+ −mV
+
V
D
dV
mnVT + V + − mVD
1 + mnVT
Again we use VD = 0.7 V.
(c)
(i)
dvo
2 × 25.2 × 10−3
=
= 0.0054
dV +
2 × 25.2 × 10−3 + 10 − 0.7
(ii)
3 × 2 × 25.2 × 10−3
dvo
=
= 0.019
dV +
3 × 2 × 25.2 × 10−3 + 10 − 3 × 0.7
3.61 Design a diode voltage regulator to supply 1.5 V to a 150 Ω load. Use two
diodes specified to have a 0.7 V drop at a current of 10 mA and n = 1. The
diodes are to be connected to a +5 V supply through a resistor R. Specify the
value of R. What is the diode current with the load connected? What is the
increase resulting in the output voltage when the load is disconnected? What
change results if the load resistance is reduced to 100 Ω? To 75 Ω? To 50 Ω?
First we compute IS from the 10 mA point as
IS =
iD
vD
nVT
=
10 × 10−3
0.7
25.2×10−3
= 8.64 × 10−15 A
e
e
Next compute the amount of current which is required to produce a 0.75 V drop across one
diode.
vD
0.75
iD = IS e VT = 8.64 × 10−15 e 25.2×10−3 = 72.8 mA
We have this current through the resistor, as well as the curent through the 150 Ω load
resistor. The current through the load is
vL
1.5
=
= 10 mA
RL
150
and the size of the resistor can then be found from
iL =
V = R (iD + iL ) + 2vD
5 − 1.5
V − 2vD
=
= 42.3 Ω
iD + iL
72.8 + 10
If the load is disconnected let’s assume that the additional small 10 mA goes through the
diodes. Let’s compute the diode resistance, rD ,
R=
rD =
diD
dvD
−1
iD =72.8 mA
=
ID
nVT
The change in voltage is then
5
=
nVT
25.2 × 10−3
=
= 0.35 Ω
ID
72.8 × 10−3
∆vO = ∆iD rD
If the load is disconnected, ∆iD = 10 mA, and ∆vD = 10 × 10−3 × 0.35 = 3.5 mV
1.5
1.5
− 100
= −5 mA, and then ∆vD =
If the load resistance is reduced to 100 Ω, ∆iD = 150
−3
−5 × 10 × 0.35 = −1.8 mV.
1.5
If the load resistance is reduced to 75 Ω, ∆iD = 150
− 1.5
= −10 mA, and then ∆vD =
75
−3.5 mV.
1.5
If the load resistance is reduced to 50 Ω, ∆iD = 150
− 1.5
= −20 mA, and then ∆vD =
50
−7 mV.
6